Week 2 – part 4: Threshold in the Hodgkin-Huxley Model · The Hodgkin-Huxley model Wulfram...
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Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 2 – Biophysical modeling: The Hodgkin-Huxley model Wulfram Gerstner EPFL, Lausanne, Switzerland 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels Week 2 – part 4: Threshold in the Hodgkin-Huxley Model
Transcript of Week 2 – part 4: Threshold in the Hodgkin-Huxley Model · The Hodgkin-Huxley model Wulfram...
Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 2 – Biophysical modeling: The Hodgkin-Huxley model
Wulfram Gerstner EPFL, Lausanne, Switzerland
2.1 Biophysics of neurons - Overview 2.2 Reversal potential
- Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 4: Threshold in the Hodgkin-Huxley Model
Neuronal Dynamics – 2.4. Threshold in HH model inside
outside
Ka
Na
Ion channels Ion pump
)()()()( 43 tIEugEungEuhmgdtduC llKKNaNa +−−−−−−=
leakI
)()(0
uumm
dtdm
mτ−
−=
)()(0
uuhh
dtdh
hτ−
−=
inside
outside
Ka
Na
Ion channels Ion pump
u u
h0(u) m0(u) )(uhτ
)(umτ
C gl gK gNa
I
Where is the threshold for firing?
Neuronal Dynamics – 2.4. Threshold in HH model
NaI KI
Constant current input I0
C gl gK gNa
I
Threshold? for repetitive firing
(current threshold)
Neuronal Dynamics – 2.4. Threshold in HH model
inside
outside
Ka
Na
Ion channels Ion pump
pulse input I(t)
Threshold?
- AP if amplitude 7.0 units - No AP if amplitude 6.9 units
(pulse with 1ms duration)
(and pulse with 0.5 ms duration?)
Neuronal Dynamics – 2.4. Threshold in HH model
)()()()( 43 tIEugEungEuhmgdtduC llKKNaNa +−−−−−−=
)()(0
uumm
dtdm
mτ−
−=
)()(0
uuhh
dtdh
hτ−
−=
Stim. NaI KI leakI
u u
h0(u) m0(u) )(uhτ
)(umτ
pulse input I(t)
Mathematical explanation
Neuronal Dynamics – 2.4. Threshold in HH model
3 ( ) ( )Na Na K leakduC g m h u E I I I tdt
= − − − − +
)()(0
uumm
dtdm
mτ−
−=
)()(0
uuhh
dtdh
hτ−
−=
Stim. NaIu u
h0(u) m0(u) )(uhτ
)(umτ
pulse input I(t)
Neuronal Dynamics – 2.4. Threshold in HH model
)()()()( 43 tIEugEungEuhmgdtduC llKKNaNa +−−−−−−=
)()(0
uumm
dtdm
mτ−
−=
)()(0
uuhh
dtdh
hτ−
−=
Stim. NaI KI leakI
u u
h0(u) m0(u) )(uhτ
)(umτ
pulse input I(t)
Neuronal Dynamics – 2.4. Threshold in HH model
0 ( )( )n
n n udndt uτ
−= −
Why start the explanation with m and not h?
What about n?
Where is the threshold?
Neuronal Dynamics – 2.4. Threshold in HH model
3
4
( )
( )( )( )
Na Na
K K
l l
duC g m h u Edt
g n u Eg u EI t
= − −
− −
− −
+
Neuronal Dynamics – 2.4. Threshold in HH model
There is no strict threshold: Coupled differential equations
‘Effective’ threshold in simulations?
0 20 ms
Strong stimulus
Action potential 100
mV
0 ϑ
strong stimuli
refractoriness
Strong stimulus
100
mV
0
Where is the firing threshold?
Refractoriness! Harder to elicit a second spike
Neuronal Dynamics – 2.4. Refractoriness in HH model
I(t)
100
mV
0
Stimulation with time-dependent input current
Neuronal Dynamics – 2.4. Simulations of the HH model
I(t) 0
5
-5
mV
mV
0
100 100
mV
0
Subthreshold response Spike
Neuronal Dynamics – 2.4. Simulations of the HH model
Step current input I2 I
Neuronal Dynamics – 2.4. Threshold in HH model
step input I2 I
Where is the firing threshold?
pulse input I(t)
ramp input
There is no threshold - no current threshold - no voltage threshold
‘effective’ threshold - depends on typical input
...)(3 −−−= NaNa EuhmgdtduC
Neuronal Dynamics – 2.4. Threshold in HH model
Response at firing threshold?
ramp input/ constant input
I0
Type I type II
I0 I0
f f f-I curve f-I curve
Neuronal Dynamics – 2.4. Type I and Type II Hodgkin-Huxley model with standard parameters (giant axon of squid)
Hodgkin-Huxley model with other parameters (e.g. for cortical pyramidal Neuron )
Neuronal Dynamics – 2.4. Hodgkin-Huxley model - 4 differential equations - no explicit threshold - effective threshold depends on stimulus - BUT: voltage threshold good approximation
Giant axon of the squid à cortical neurons - Change of parameters - More ion channels - Same framework
Exercise – 2. 4. Hodgkin-Huxley model – ion channel
)()(4 tIEungdtduC KK +−−=
stimulus
inside
outside
Ka
Na
Ion channels Ion pump
C gK
I
u u
n0(u)
)(unτ
Determine ion channel dynamics
adapted from Hodgkin&Huxley 1952
apply voltage step
voltage step u2 u
Exercise – 2. 4. Hodgkin-Huxley model – ion channel
)()(4 tIEungdtduC KK +−−=
stimulus
inside
outside
Ka
Na
Ion channels Ion pump C gK I
u u
n0(u)
)(unτ
Determine ion channel dynamics
adapted from Hodgkin&Huxley 1952
apply voltage step
